## American Go Association Rules - Explanation

This document contains an explanation of the official rules of Go for the AGA (April 1991).

See also an introduction and the rules with commentary and the shorter rules.

## Appendix: Demonstration of the Relationship of Area and Territory Scoring

This appendix shows that with normal play, under the given rules the "area" and "territory" scores of a game will always be the same.

For the sake of simplicity, start by assuming an even game, that both players play exactly the same number of stones, only passing at the end, and that at the end of the game there are no neutral points (all dame filled and no seki). Make the following definitions:

```        Sw =   number of stones played by white
Sb  =   number of stones played by black
Pw =   number of white prisoners
Pb  =   number of black prisoners
Aw =   number of white stones on the board
Ab  =   number of black stones on the board
Tw =   number of points of territory surrounded by white
Tb  =   number of points of territory surrounded by black
```
Now, we have the following relationships. Since we are assuming that both players have played exactly the same number of stones, we have:
```                Ab + Pb  =   Sb  =  Sw =  Aw + Pw
```
This means that:
```                Ab  -  Aw = Pw  -   Pb
```
So, adding Tb - Tw to both sides,
```                (Ab + Tb) - (Aw + Tw) = (Tb -  Pb) - (Tw - Pw)          [1]
```
But the left-hand side of this expression is precisely the score according to area counting, while the right-hand side is the score according to territorial counting!

If a player passes prior to the end of the game, it will reduce that player's area score by one point per pass relative to the corresponding territorial score. The convention of handing a "pass" stone to the opponent when passing keeps the two scores equal. (In general, it can only hurt a player to pass prior to the end of the game.)

There are also certain rare situations where a game ends with "one-sided dame" (see Figure 5) which one side can fill but the other cannot. Each additional stone played represents the gain of a point under area counting. But since the opponent will be forced to hand over a "pass" stone on his or her move, each additional stone played also represents the gain of a point under territory counting--the two remain equivalent!

Finally, in a handicap game, the additional points of compensation paid by Black to White can be thought of as "reverse pass stones" ensuring that both players have, in effect, still played exactly the same number of stones.

If we assume that the two players have played the same number of stones, with no neutral points left on the board, and that the score in equation [1] is equal to k, we have:

```                (Ab + Tb) - (Aw + Tw)  =  k
```
But
```                (Ab + Tb) + (Aw + Tw) = 361                                 [2]
```
So k must be odd!

This implies that if such a game is even on the board by traditional territorial counting (without pass stones), Black must have made the last move ! At one time, the Chinese rules compensated White with an extra point when Black got the last move. If Black's last move was to fill a ko he or she had won, however, it was deemed unfair to penalize him or her, so eventually the Chinese removed this proviso. Requiring that White always have the last move and using pass stones removes the possibility of a "pass fight" over who gets the very last move.

If there are neutral points on the board at the end of the game (presumably in seki, since the players would naturally fill all dame under the area system), the same argument still shows that the two systems give the same result if the players have played the same number of stones, but the parity of k will depend on the number of neutral points; if there are an odd number of neutral points, k will be even, and vice versa. This may explain why some rule systems go to great lengths to award all points in seki.

Finally, note that in the confirmation phase, by our rules, the final result remains the same (that is, the "same" as would be calculated before playing out the confirmation phase if the status of all groups were taken to be whatever it proves to be through the confirmation process!) Since the game is over, we can assume that all empty points belong to the territory of one or the other of the players. Under area counting, stones of either colour played into one's own territory or into the opponent's territory will not change the score--nor will the "pass" stones. Under territorial counting, every stone played into one's own or the opponent's territory will cost a point--but by requiring that the players make the same number of moves, and by insuring that even passes cost a point (the "pass" stones), we insure that the end result is still the same.

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